Local convergence of simultaneous min-max algorithms to differential equilibrium on Riemannian manifold

We study min-max algorithms to solve zero-sum differential games onRiemannian manifold.Based on the notions ofdifferential Stackelberg equilibriumand differential Nash equilibrium on Riemannian manifold,we analyze the local convergence of two representative deterministic simultaneous algorithms $\tau$-GDA and $\tau$-SGAto such equilibria.Sufficient conditions are obtained to establish the linear convergence rateof $\tau$-GDA based on the Ostrowski theorem on manifold and spectral analysis. To avoid strong rotational dynamics in $\tau$-GDA, $\tau$-SGA is extended fromthe symplectic gradient-adjustment method in Euclidean space.We analyze an asymptotic approximation of $\tau$-SGA when the learning rate ratio $\tau$ is big. In some cases, it can achieve a faster convergence rate to differential Stackelberg equilibrium compared to $\tau$-GDA. We show numerically how the insights obtained from theconvergence analysis may improvethe training of orthogonal Wasserstein GANs using stochastic $\tau$-GDA and $\tau$-SGA on simple benchmarks.